Probabilistic circuits (PCs) are a powerful modeling framework for representing tractable probability distributions over combinatorial spaces. In machine learning and probabilistic programming, one is often interested in understanding whether the distributions learned using PCs are close to the desired distribution. Thus, given two probabilistic circuits, a fundamental problem of interest is to determine whether their distributions are close to each other. The primary contribution of this paper is a closeness test for PCs with respect to the total variation distance metric. Our algorithm utilizes two common PC queries, counting and sampling. In particular, we provide a poly-time probabilistic algorithm to check the closeness of two PCs when the PCs support tractable approximate counting and sampling. We demonstrate the practical efficiency of our algorithmic framework via a detailed experimental evaluation of a prototype implementation against a set of 475 PC benchmarks. We find that our test correctly decides the closeness of all 475 PCs within 3600 seconds.
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Speech enhancement is a critical component of many user-oriented audio applications, yet current systems still suffer from distorted and unnatural outputs. While generative models have shown strong potential in speech synthesis, they are still lagging behind in speech enhancement. This work leverages recent advances in diffusion probabilistic models, and proposes a novel speech enhancement algorithm that incorporates characteristics of the observed noisy speech signal into the diffusion and reverse processes. More specifically, we propose a generalized formulation of the diffusion probabilistic model named conditional diffusion probabilistic model that, in its reverse process, can adapt to non-Gaussian real noises in the estimated speech signal. In our experiments, we demonstrate strong performance of the proposed approach compared to representative generative models, and investigate the generalization capability of our models to other datasets with noise characteristics unseen during training.
The present study defines a new statistic for detecting laboratory effects in the analysis of ordinal variation (ORDANOVA). The ORDANOVA is an analysis method similar to one-way analysis of variance for analysing ordinal data obtained from interlaboratory comparison studies. In this paper, we present an approximate continuous distribution for the new statistic for the case of an arbitrary number of ordinal levels, and we demonstrate that $alpha$-percentiles of the distribution are suitable criteria for conducting statistical tests. In addition, a real example involving data from an interlaboratory comparison study is analysed using the proposed statistic.
The optimal design of magnetic devices becomes intractable using current computational methods when the number of design parameters is high. The emerging physics-informed deep learning framework has the potential to alleviate this curse of dimensionality. The objective of this paper is to investigate the ability of physics-informed neural networks to learn the magnetic field response as a function of design parameters in the context of a two-dimensional (2-D) magnetostatic problem. Our approach is as follows. We derive the variational principle for 2-D parametric magnetostatic problems, and prove the existence and uniqueness of the solution that satisfies the equations of the governing physics, i.e., Maxwell's equations. We use a deep neural network (DNN) to represent the magnetic field as a function of space and a total of ten parameters that describe geometric features and operating point conditions. We train the DNN by minimizing the physics-informed loss function using a variant of stochastic gradient descent. Subsequently, we conduct systematic numerical studies using a parametric EI-core electromagnet problem. In these studies, we vary the DNN architecture trying more than one hundred different possibilities. For each study, we evaluate the accuracy of the DNN by comparing its predictions to those of finite element analysis. In an exhaustive non-parametric study, we observe that sufficiently parameterized dense networks result in relative errors of less than 1%. Residual connections always improve relative errors for the same number of training iterations. Also, we observe that Fourier encoding features aligned with the device geometry do improve the rate of convergence, albeit higher-order harmonics are not necessary. Finally, we demonstrate our approach on a ten-dimensional problem with parameterized geometry.
Probabilistic databases (PDBs) model uncertainty in data in a quantitative way. In the established formal framework, probabilistic (relational) databases are finite probability spaces over relational database instances. This finiteness can clash with intuitive query behavior (Ceylan et al., KR 2016), and with application scenarios that are better modeled by continuous probability distributions (Dalvi et al., CACM 2009). We formally introduced infinite PDBs in (Grohe and Lindner, PODS 2019) with a primary focus on countably infinite spaces. However, an extension beyond countable probability spaces raises nontrivial foundational issues concerned with the measurability of events and queries and ultimately with the question whether queries have a well-defined semantics. We argue that finite point processes are an appropriate model from probability theory for dealing with general probabilistic databases. This allows us to construct suitable (uncountable) probability spaces of database instances in a systematic way. Our main technical results are measurability statements for relational algebra queries as well as aggregate queries and Datalog queries.
Spatial statistics is concerned with the analysis of data that have spatial locations associated with them, and those locations are used to model statistical dependence between the data. The spatial data are treated as a single realisation from a probability model that encodes the dependence through both fixed effects and random effects, where randomness is manifest in the underlying spatial process and in the noisy, incomplete, measurement process. The focus of this review article is on the use of basis functions to provide an extremely flexible and computationally efficient way to model spatial processes that are possibly highly non-stationary. Several examples of basis-function models are provided to illustrate how they are used in Gaussian, non-Gaussian, multivariate, and spatio-temporal settings, with applications in geophysics. Our aim is to emphasise the versatility of these spatial statistical models and to demonstrate that they are now centre-stage in a number of application domains. The review concludes with a discussion and illustration of software currently available to fit spatial-basis-function models and implement spatial-statistical prediction.
Personalized recommender systems are playing an increasingly important role as more content and services become available and users struggle to identify what might interest them. Although matrix factorization and deep learning based methods have proved effective in user preference modeling, they violate the triangle inequality and fail to capture fine-grained preference information. To tackle this, we develop a distance-based recommendation model with several novel aspects: (i) each user and item are parameterized by Gaussian distributions to capture the learning uncertainties; (ii) an adaptive margin generation scheme is proposed to generate the margins regarding different training triplets; (iii) explicit user-user/item-item similarity modeling is incorporated in the objective function. The Wasserstein distance is employed to determine preferences because it obeys the triangle inequality and can measure the distance between probabilistic distributions. Via a comparison using five real-world datasets with state-of-the-art methods, the proposed model outperforms the best existing models by 4-22% in terms of recall@K on Top-K recommendation.
Markov Logic Networks (MLNs), which elegantly combine logic rules and probabilistic graphical models, can be used to address many knowledge graph problems. However, inference in MLN is computationally intensive, making the industrial-scale application of MLN very difficult. In recent years, graph neural networks (GNNs) have emerged as efficient and effective tools for large-scale graph problems. Nevertheless, GNNs do not explicitly incorporate prior logic rules into the models, and may require many labeled examples for a target task. In this paper, we explore the combination of MLNs and GNNs, and use graph neural networks for variational inference in MLN. We propose a GNN variant, named ExpressGNN, which strikes a nice balance between the representation power and the simplicity of the model. Our extensive experiments on several benchmark datasets demonstrate that ExpressGNN leads to effective and efficient probabilistic logic reasoning.
Knowledge graph reasoning, which aims at predicting the missing facts through reasoning with the observed facts, is critical to many applications. Such a problem has been widely explored by traditional logic rule-based approaches and recent knowledge graph embedding methods. A principled logic rule-based approach is the Markov Logic Network (MLN), which is able to leverage domain knowledge with first-order logic and meanwhile handle their uncertainty. However, the inference of MLNs is usually very difficult due to the complicated graph structures. Different from MLNs, knowledge graph embedding methods (e.g. TransE, DistMult) learn effective entity and relation embeddings for reasoning, which are much more effective and efficient. However, they are unable to leverage domain knowledge. In this paper, we propose the probabilistic Logic Neural Network (pLogicNet), which combines the advantages of both methods. A pLogicNet defines the joint distribution of all possible triplets by using a Markov logic network with first-order logic, which can be efficiently optimized with the variational EM algorithm. In the E-step, a knowledge graph embedding model is used for inferring the missing triplets, while in the M-step, the weights of logic rules are updated based on both the observed and predicted triplets. Experiments on multiple knowledge graphs prove the effectiveness of pLogicNet over many competitive baselines.
This paper studies the problem of domain division problem which aims to segment instances drawn from different probabilistic distributions. Such a problem exists in many previous recognition tasks, such as Open Set Learning (OSL) and Generalized Zero-Shot Learning (G-ZSL), where the testing instances come from either seen or novel/unseen classes of different probabilistic distributions. Previous works focused on either only calibrating the confident prediction of classifiers of seen classes (W-SVM), or taking unseen classes as outliers. In contrast, this paper proposes a probabilistic way of directly estimating and fine-tuning the decision boundary between seen and novel/unseen classes. In particular, we propose a domain division algorithm of learning to split the testing instances into known, unknown and uncertain domains, and then conduct recognize tasks in each domain. Two statistical tools, namely, bootstrapping and Kolmogorov-Smirnov (K-S) Test, for the first time, are introduced to discover and fine-tune the decision boundary of each domain. Critically, the uncertain domain is newly introduced in our framework to adopt those instances whose domain cannot be predicted confidently. Extensive experiments demonstrate that our approach achieved the state-of-the-art performance on OSL and G-ZSL benchmarks.
We present an implementation of a probabilistic first-order logic called TensorLog, in which classes of logical queries are compiled into differentiable functions in a neural-network infrastructure such as Tensorflow or Theano. This leads to a close integration of probabilistic logical reasoning with deep-learning infrastructure: in particular, it enables high-performance deep learning frameworks to be used for tuning the parameters of a probabilistic logic. Experimental results show that TensorLog scales to problems involving hundreds of thousands of knowledge-base triples and tens of thousands of examples.
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